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Solving Equations
Introductory example
.
Solve the equation
=
x
(
)
+
x
2
15
 (i).
We need to find all the real numbers
x
such the statement
=
x
(
)
+
x
2
15
is true.
Putting the problem into words, we need to find a number such that the product of this number with a second number, that is obtained by
adding 2 to the first number, is 15. Thinking just in terms of integers (whole numbers) we can quickly find the solution
=
x
3
by a "trial
and error" approach. However, there is another solution.
Rewriting the equation in the form
=
+
x
2
2
x
15
by expanding the left side using distributivity of multiplication over addition.
Then we can rewrite the equation in the form
=
+

x
2
2
x
15
0.
so that the left side can be factored to give
=
(
)

x
3 (
)
+
x
5
0
 (ii).
At this stage we can use the fact that a product of two real numbers
a
and
b
is equal to zero exactly when either one or other of the two
numbers
a
and
b
is equal to zero, that is,
=
a b
0
exactly when
=
a
0
or
=
b
0.
________________
This is equivalent to the following two facts taken together.
•
Multiplying any real number by 0 gives 0, that is,
=
0
. a
0
for any real number
a
.
•
Multiplying two nonzero numbers together gives a product that is not zero, that is, if
≠
a
0
and
≠
b
0, then
≠
a b
0.
We apply this "zero product principle"
to equation (ii) to see that
=
(
)

x
3 (
)
+
x
5
0 exactly when either
=

x
3
0
or
=
+
x
5
0, which
means the same as saying
=
x
3
or
=
x

5.
Terminology used in connection with solving equations
.
If we introduce the concept of the
solution set
of an equation as the set of all solutions, then the solution set of equation (i) is
{
,

5 3},
that is, the solution set is
{
x

x
is a real number and
=
x
(
)
+
x
2
15 } = {
,

5 3}.
In general, two equations are said to be
equivalent
exactly when they have the same solution set. The process of solving an equation
(usually) involves writing down a sequence of equivalent equations, ending up with an equation from which the solutions can readily be
obtained. The logical connective <=> which means "is equivalent to" may be used here. Thus the solution of equation (i) can be written
down as follows.
=
x
(
)
+
x
2
15
<=>
=
+
x
2
2
x
15
<=>
=


x
2
2
x
15
0
<=>
=
(
)

x
3 (
)
+
x
5
0
<=>
=
x
3
or
=
x

5.
_______
Each occurrence of
<=> can be read as either "is equivalent to" or "which is equivalent to".
One can then finish by stating that the solution set of the original equation is
{ ,
3

5}.
The symbol
<=> is often omitted, however.
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View Full DocumentProcesses that
produce an
equivalent equation when applied to an equation involving a single variable or
"unknown" x
.
•
Adding
a constant or an algebraic expression involving the variable
x
to both sides of an equation provided that the expression added
can be evaluated for all real numbers.
( More generally, it is sufficient for the expression that is to be added to both sides to have a real number value for each of the
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 Spring '08
 Uri
 Real Numbers, Equations

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